Corrigendum to “Convergence of adaptive, discontinuous Galerkin methods”

Abstract:

The first statement of Lemma 11 in our recent paper [KG18] (Math. Comp. 87 (2018), no. 314, 2611–2640) is incorrect: For the sequence $\{\mathcal {G}_{k}\}_k$ of nested admissible partitions produced by the adaptive discontinuous Galerkin method (ADGM) we have $\mathcal {G}^+\coloneq \bigcup _{k\ge 0}\bigcap _{j\ge k}\mathcal {G}_{j}$, and $\Omega ^+\coloneq \operatorname {interior}\left (\bigcup \{ E: E\in \mathcal {G}^+\}\right )$. In the first line of the proof of [KG18, Lemma 11 on p. 2620], we used that \begin{align*} |\Omega |=|\operatorname {interior}(\Omega \setminus \Omega ^+)|+|\Omega ^+|, \end{align*} where $|\cdot |$ denotes the Lebesgue measure. This, however, is not true in general, since there are counter examples where $\Omega ^+$ is dense in $\Omega$ and \[ 0=|\operatorname {interior}(\Omega \setminus \Omega ^+)|<|\Omega \setminus \Omega ^+|. \]

Below, we present the required minor modifications to complete the proof of the main result stating convergence of the ADGM of [KG18] and address some typos regarding the broken dG-norm. A corrected full version of the article is available at arXiv:1909.12665v2.